Monadic theory of term rewritings

The monadic second-order theory of term rewritings is considered. It is shown that the monadic theory of the rewriting (or the suffix rewriting) of a ground rewrite system is undecidable. Furthermore, the first-order theory is undecidable for the prefix derivation according to a linear context-free grammar on linear terms. Nevertheless, a new notion on terms with variables is introduced: a term is entire if each of its subterms either is a variable, or is without variable or has the same variables as the term. It is shown that the monadic theory is decidable (respectively undecidable) for the prefix rewriting according to a rewrite system on entire terms, with an axiom (respectively without axiom).<>